Evaluate the following integrals:
`int (e^(x))/(a^(x))dx`

Answer

To evaluate the integral \( \int \frac{e^x}{a^x} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integrand. Notice that \( a^x \) can be expressed as \( e^{x \ln a} \). Thus, we can rewrite the integral as:
\[ \int \frac{e^x}{a^x} \, dx = \int \frac{e^x}{e^{x \ln a}} \, dx = \int e^{x(1 - \ln a)} \, dx \]

### Step 2: Apply the Integral Formula
Now we can apply the formula for the integral of an exponential function:
\[ \int e^{kx} \, dx = \frac{1}{k} e^{kx} + C \]
where \( k = 1 - \ln a \). Therefore, we have:
\[ \int e^{x(1 - \ln a)} \, dx = \frac{1}{1 - \ln a} e^{x(1 - \ln a)} + C \]

### Step 3: Substitute Back
Now we substitute back to express the answer in terms of \( a \):
\[ = \frac{1}{1 - \ln a} e^{x(1 - \ln a)} + C = \frac{1}{1 - \ln a} \cdot \frac{e^x}{a^x} + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{e^x}{a^x} \, dx = \frac{e^x}{a^x (1 - \ln a)} + C \]